91Ó°ÊÓ

The cosine function, represented as \(\cos(\theta)\), is a fundamental trigonometric function that maps angles to a value between -1 and 1. It is an even function, meaning that \(\cos(-\theta) = \cos(\theta)\). This symmetry is around the vertical axis, making it an important property when solving trigonometric equations. The graph of the cosine function is a wave that repeats every \(2\pi\) radians. This makes it ideal for describing periodic phenomena, such as sound waves and seasonal patterns.

When solving equations like \(\cos(2x) = -\frac{1}{2}\), we use properties of the cosine function. Here, we specifically look for angles where the cosine value is \(-\frac{1}{2}\), and this can help us find all possible solutions within a given range. The cosine of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) are both \(-\frac{1}{2}\), illustrating solutions at these key angles within a complete cycle.

In trigonometry, finding a general solution to an equation involves understanding the periodic nature of trigonometric functions. For instance, the cosine function is periodic with a period of \(2\pi\), meaning every \(2\pi\) units, the function repeats its values. This is critical when solving an equation like \(\cos(\theta) = -\frac{1}{2}\).

For \(\theta\), where \(\cos(\theta) \) is \(-\frac{1}{2}\), the general solution can be written as:

• \(\theta = \frac{2\pi}{3} + 2k\pi\)
• \(\theta = \frac{4\pi}{3} + 2k\pi\)

where \(k\) is an integer. This represents all the angles that will satisfy the original condition by accounting for the whole infinite set of angle solutions due to the periodic nature.

Periodic functions, like the cosine function, repeat their values over regular intervals, known as periods. The period of the cosine function is \(2\pi\) radians, meaning its values repeat every \(2\pi\) change in the angle. This repetitive pattern is essential in solving trigonometric equations, especially when we want to find solutions over a specific interval.

Understanding this concept helps us realize that if we have a solution for a trigonometric equation, such as \(x = \frac{\pi}{3}\), then \(x + 2k\pi\) will also be solutions for any integer \(k\). This periodic property thus enables us to fill in all anti-nodes of the trigonometric wave for the entire domain by understanding just one cycle.

Interval notation is a simplified way of representing continuous sets of numbers and is especially useful in describing the solutions to inequalities. In this context, after solving \(\cos 2x > -\frac{1}{2}\) graphically, we express our solutions using interval notation to concisely list the valid ranges of \(x\).

When the graph of \(y = \cos 2x\) is seen above the line \(y = -\frac{1}{2}\), we determine intervals such as:

• \((0, \frac{\pi}{3})\)
• \((\frac{2\pi}{3}, \frac{4\pi}{3})\)
• \((\frac{5\pi}{3}, 2\pi)\)

These intervals specify where the inequality is true, using round brackets to indicate that endpoints are not included. Interval notation provides a precise, concise way to express these ranges for anyone solving similar equations.